We provide an elementary derivation of the one-dimensional quantum harmonic oscillator propagator, using a mix of approaches, such as path integrals, canonical operators, and ladder operators. This way we take the best of each world, and find the propagator with as few tears as possible.

1.
N. A.
Lemos
,
Analytical Mechanics
(
Cambridge U. P.
,
Cambridge, UK
,
2018
).
2.
C.
Cohen-Tannoudji
,
B.
Diu
, and
F.
Laloë
,
Quantum Mechanics
(
Wiley
,
Hoboken, NJ
,
1991
), Vol. 1.
3.
K.
Gottfried
and
T. M.
Yan
,
Quantum Mechanics: Fundamentals
(
Springer
,
Heidelberg, Germany
,
2004
).
4.
F. A.
Barone
,
H.
Boschi-Filho
, and
C.
Farina
, “
Three methods for calculating the Feynman propagator
,”
Am. J. Phys.
71
,
483
491
(
2003
).
5.
B.
Holstein
, “
The harmonic oscillator propagator
,”
Am. J. Phys.
66
,
583
589
(
1998
).
6.
L.
Moriconi
, “
An elementary derivation of the harmonic oscillator propagator
,”
Am. J. Phys.
72
,
1258
1259
(
2004
).
7.
L. F.
Barrágan-Gil
and
R.
Walser
, “
Harmonic oscillator thermal density matrix: First-order differential equations for the position representation
,”
Am. J. Phys.
86
,
22
24
(
2018
).
8.
K.
Hira
, “
Derivation of the harmonic oscillator propagator using the Feynman path integral and recursive relations
,”
Eur. J. Phys.
34
,
777
785
(
2013
).
9.
J.
Shao
, “
Elementary derivation of the quantum propagator for the harmonic oscillator
,”
Am. J. Phys.
84
,
770
774
(
2016
).
10.
N.
Thornber
and
E.
Taylor
, “
Propagator for the simple harmonic oscillator
,”
Am. J. Phys.
66
,
1022
1024
(
1998
).
11.

One quick way to obtain the ground state wave function is to note that, since it does not have zeros,12 we can write it as ψ(x)=eϕ(x). Plugging this into the Schrodinger equation, we find the solution for ϕ(x) easily.

12.
M.
Moriconi
, “
Nodes of wavefunctions
,”
Am. J. Phys.
75
,
253
254
(
2007
).
13.
We consider T > 0 and avoid the use of the Heaviside function
, which guarantees causality.
14.
L. S.
Schulman
,
Techniques and Applications of Path Integration
(
Dover Publications
,
New York
,
2005
).
15.
C.
Grosche
and
F.
Steiner
,
Handbook of Path Integrals
, Springer Tracts in Modern Physics Vol.
145
(
Springer
,
Heidelberg, Germany
,
1998
).
16.

In the case of quadratic Lagrangians, S(η,η̇) is also quadratic. In general, this is not the case, but if we restrict it to the quadratic part of the new action, we obtain the semiclassical approximation.14 

17.

Since we only performed a translation in the variable x(t), the Jacobian of this transformation is 1.

18.

An alternative method to find this result is by using the generating function for the Hermite polynomials, e2xtt2=n=0Hn(x)tnn!, substitute x = 0, and compare the powers of t.

19.

This trick suppresses states for n bigger than, roughly, N1/(ϵωT).

20.
C.
Bender
,
D.
Brody
, and
M.
Parry
, “
Making sense of the divergent series for reconstructing a Hamiltonian from its eigenstates and eigenvalues
,”
Am. J. Phys.
88
,
148
152
(
2020
).
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